arXiv subjects

Shoya Motonaga

Publications and source records attributed to Shoya Motonaga.

8 recordsLinked to original sources

arXiv2025

Explicit formulae and topological descriptions of action-minimizing sets of a full shift with an uncountable alphabet $[0,1]$

We completely solve ergodic optimization of a full shift with an uncountable alphabet $[0,1]$, which is one of the most well-known examples of infinite dimensional dynamical systems with positive mean dimension (and thus with infinite topological entropy), for potentials depending only on the first two coordinates with the twist condition as well as giving explicit formulae of the associated Mather set and the Aubry set. Moreover, we investigate the total disconnectedness of the (quotient) Aubry set, in which case the differentiability of the potential function makes a crucial difference. Although these results imply that the (quotient) Aubry set is small enough, we give a complete characterization of an analogical object of the Aubry set, called the Ma\~{n}\'e set, and show that it is much larger than the Aubry set so that it contains cubes of any finite dimension. In our proofs, estimates for ``connecting orbits" play key roles, and we establish them by combining three different perspectives in our symbolic setting; weak KAM subaction approach for symbolic dynamics, Ma\~{n}\'e's formulation in Lagrangian systems, and Bangert's variational approach for twist maps.

math.DS↗

arXiv2025

The Aubry set for the XY model and typicality of periodic optimization for 2-locally constant potentials

We consider the Aubry set for the XY model, symbolic dynamics $([0,1]^{\mathbb{N}_0},σ)$ with the uncountable symbol $[0,1]$, and study its action-optimizing properties. Moreover, for a potential function that depends on the first two coordinates we obtain an explicit expression of the set of optimal periodic measures and a detailed description of the Aubry set. We also show the typicality of periodic optimization for 2-locally constant potentials with the twist condition. Our approach combines the weak KAM method for symbolic dynamics and variational techniques for twist maps.

math.DS↗

arXiv2024

Minimax aspects of optimizations in ergodic theory

We study optimization problems in ergodic theory from the view point of minimax problems. We give minimax characterizations of maximum ergodic averages involving time averages. Our approach works for the abstract variational principle of generalized pressure functions which is proved by Biś et al. (2022). We also describe the relationship between our minimax results and the Fenchel-Rockafellar duality.

math.DS↗

arXiv2024

Constrained Ergodic optimization for generic continuous functions

One of the fundamental results of ergodic optimization asserts that for any dynamical system on a compact metric space with the specification property and for a generic continuous function $f$ every invariant probability measure that maximizes the space average of $f$ must have zero entropy. We establish the analogical result in the context of constraint ergodic optimization, which is introduced by Garibaldi and Lopes (2007).

math.DS↗

arXiv2022

Properties of action-minimizing sets and weak KAM solutions via Mather's averaging functions

We study properties of action-minimizing invariant sets for Tonelli Lagrangian and Hamiltonian systems and weak KAM solutions to the Hamilton-Jacobi equation in terms of Mather's averaging functions. Our principal discovery is that exposed points and extreme points of Mather's alpha function are closely related to disjoint properties and graph properties of the action-minimizing invariant sets, which is also related to $C^0$ integrability of the systems and the existence of smooth weak KAM solutions to the Hamilton-Jacobi equation.

math.DS↗

arXiv2022

Nonintegrability of Forced Nonlinear Oscillators

In recent papers by the authors (S.~Motonaga and K.~Yagasaki, Obstructions to integrability of nearly integrable dynamical systems near regular level sets, submitted for publication, and K.~Yagasaki, Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits, submitted for publication), two different techniques which allow us to prove the real-analytic or complex-meromorphic nonintegrability of forced nonlinear oscillators having the form of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems were provided. Here the concept of nonintegrability in the Bogoyavlenskij sense is adopted and the first integrals and commutative vector fields are also required to depend real-analytically or complex-meromorphically on the small parameter. In this paper we review the theories and continue to demonstrate their usefulness. In particular, we consider the periodically forced damped pendulum and prove its nonintegrability in the above meaning.

math.DS↗

arXiv2021

Obstructions to Integrability of Nearly Integrable Dynamical Systems near Regular Level Sets

We study the existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems in the sense of Bogoyavlenskij including non-Hamiltonian ones. We especially assume that there exists a family of periodic orbits on a regular level set of the first integrals having a connected and compact component and give sufficient conditions for nonexistence of the same number of real-analytic first integrals in the perturbed systems as the unperturbed ones and for their real-analytic nonintegrability near the level set such that the first integrals and commutative vector fields depend analytically on the small parameter. We compare our results with classical results of Poincaré and Kozlov for systems written in action and angle coordinates and discuss their relationships with the subharmonic and homoclinic Melnikov methods for periodic perturbations of single-degree-of-freeedom Hamiltonian systems. We illustrate our theory for three examples containing the periodically forced Duffing oscillator.

math.DS↗

arXiv2021

Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems

We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter $\varepsilon>0$ and give several necessary conditions for their persistence. Here we treat homoclinic orbits not only to equilibria but also to periodic orbits. We also discuss some relationships of these results with the standard subharmonic and homoclinic Melnikov methods for time-periodic perturbations of single-degree-of-freedom Hamiltonian systems, and with another version of the homoclinic Melnikov method for autonomous perturbations of multi-degree-of-freedom Hamiltonian systems. In particular, we show that a first integral which converges to the Hamiltonian or another first integral as the perturbation tends to zero does not exist near the unperturbed periodic or homoclinic orbits in the perturbed systems if the subharmonic or homoclinic Melnikov functions are not identically zero on connected open sets. We illustrate our theory for four examples: The periodically forced Duffing oscillator, two identical pendula coupled with a harmonic oscillator, a periodically forced rigid body and a three-mode truncation of a buckled beam.

math.DS↗